**Published at : ** 20 Dec 2021

**Volume :** **IJtech**
Vol 12, No 6 (2021)

**DOI :** https://doi.org/10.14716/ijtech.v12i6.5210

Maknun, I.J., Zarfatina, S., 2021. Shear Correction Factor Effects on Functionally Graded Materials (FGMs) Beams using Discrete Shear Gap (DSG) Element.

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Imam Jauhari Maknun | Department of Civil and Environmental Engineering, Faculty of Engineering, Universitas Indonesia, Kampus UI Depok, Depok 16424, Indonesia |

Salfa Zarfatina | Department of Civil and Environmental Engineering, Faculty of Engineering, Universitas Indonesia, Kampus UI Depok, Depok 16424, Indonesia |

Abstract

Materials
and computational methods are essential to support the development of
infrastructure. Composite material has been used in many applications; in the
laminated composite, failure due to excessive interlaminar stresses between two
materials causes delamination. Thus, functionally graded materials (FGMs) have
emerged. A numerical computation such as the finite element method (FEM) is
widely used to support the analysis of FGMs in structural applications. The
discrete shear gap DSG element is developed using Timoshenko beam theory, where
the shear correction factor is used in their formulation. The shear correction
factor is assumed to be constant in many applications; thus, it is valid for
isotropic homogenous material. However, the effect of shear deformation
significantly impacts the results of the FGMs beam, so the shear correction
factor cannot be considered constant. Therefore, this paper presents the shear correction factor effect on static analysis of
FGMs beam using DSG element. Various boundary conditions with length thickness
ratio (*L/h *= 4) are evaluated. The DSG element yields good results in
FGMs beam for different power-law index ratios. Furthermore, the DSG element
result shows that the higher the modulus of elasticity ratio of the top-to-bottom
material, the further the difference between *k* FGMs and *k* = 5/6
(constant). The DSG element can provide precise results without shear locking.

Composite; DSG; FEM; FGMs; Shear correction factors

Introduction

Composites are the combination of materials selected based on the variety of the physical properties of each constituent material. They are combined to produce a new material with unique properties compared to those of the primary material. The application of composite material in beam structures can be found in Benbouras et al. (2017). Delamination remains a large problem in laminated composites (Reddy, 2006). In an attempt to solve this problem, new composite materials, called Functionally Graded Materials (FGMs), were first proposed in 1984 by a group of materials scientists in Japan.

The material properties vary gradually and continuously from one surface to another according to the given function. The gradation of material properties through thickness avoids sudden changes in stress distribution. In general, this material is a mixture of ceramics on the top and metal at the bottom. The main benefit of using this material is that it can withstand extreme situations like a high-temperature environment. The use of this material increases rapidly in many applications. Numerical analysis such as FEM is desired to support the development of FGMs.

FEM is one of the numerical
techniques used to support structural analysis. Many papers deal with the
recent development of FEM in plate and shell structures for isotropic material (Katili et al., 2014; Maknun et al., 2016; Irpanni et al.,
2017; Katili et al., 2017; Katili et al.,
2018a; Katili et al., 2018b; Katili et al., 2019a). The results of FEM
for the composite material can be found in (Katili
et al., 2015; Maknun et al., 2015; Katili et al., 2018c; Katili et al., 2018d;
Katili et al., 2019b; Maknun et al., 2020). From the results, FEM can
support the analysis of plate and shell structures in isotropic and composite
materials with accurate results.

FEM performs very well in
calculating FGMs beam (Simsek, 2009; Thai and Vo
2012; Nguyen et al., 2013, Aghazadeh et al.,
2014; Jing et al., 2016; Katili and Katili 2020).
Many beam elements have been proposed for application in beam structures using
either Euler Bernoulli's or Timoshenko’s (Timoshenko,
1921; Timoshenko, 1922) beam theories. The beam element developed using
Euler Bernoulli's beam theory needs C^{1} continuity and can only be
used for thin beam structures. Timoshenko beam elements were proposed to
overcome these limitations. Beam elements developed from Timoshenko beam theory
can be used for thin to thick beam structures and need only C^{0}
continuity. However, the shear locking phenomenon exists in Timoshenko beam
elements. There are several methods to overcome this phenomenon. One of these is the discrete shear gap (DSG)
method proposed by Bletzinger et al. (2000).

For the beam element developed from Timoshenko beam theory, such as the
DSG element, the shear correction factor is one of the necessary terms. This
factor obtained from the shear strain energy from equilibrium equations equated
to the shear strain energy obtained from the constitutive equation (Meena et al., 2012). In isotropic rectangular
beams, this factor is equal to *k* = 5/6. For many applications of FGMs in
beam structures, the shear correction factor is assumed to be constant. This
factor is not constant for non-isotropic material like FGMs. This factor has a
significant effect when analyzing thick beam problems in which the shear energy
is crucial, which many papers have dealt with. (Timoshenko,
1922; Nguyen et al., 2008; Hosseini-Hashem et al. 2010).

This paper aims to study the effect of the
shear correction factor on the FGMs beam by using the DSG element. The equation
for the shear correction factor is taken from Meena
et al. (2012). FGMs beams with various boundary
conditions and length thickness ratios (*L/h *= 4) are evaluated. Different power-law indexes (*n*)
are also assessed to understand the convergence behavior of the DSG element in
the FGMs beam.

Conclusion

From
the analysis, it can be concluded that the performance of DSG elements on FGMs
beams yields good results. The
convergence results for *k *= 5/6 and *k* FGMs are close to the HOSDT reference
for the number of elements ? 16. The top and bottom elastic modulus (*Et/Eb*)
ratio and the power-law index (*n*) influence the shear correction factor.
The greater the *Et*/*Eb *ratio*,* the greater the effect of the shear correction factor. In the
case with *Et*/*Eb *= 0.35 for thick beams (*L/h
*= 4), the error of displacement ranged from 0.184% (clamped-free) to 1.394%
(clamped-clamped). In the case with *Et*/*Eb *= 20 for thick beams (*L/h *= 4),
the error of displacement ranged from 4.432% (clamped-free) to 25.751%.
(clamped-clamped). The largest error occurred in the clamped-clamped support
shear energy is more important here than in other supports.

Acknowledgement

The
financial support from Penelitian Dasar Unggulan Perguruan Tinggi 2021 Nomor:
NKB-199/UN2.RST/HKP.05.00/2021 is gratefully acknowledged.

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